We are developing X-ray microcalorimeters based on Ti/Au transition-edge sensors (TES). Among sensors we have fabricated, one with a Cu absorber at the center of the TES shows a particularly good X-ray energy resolution: 1.56 eV at 250 eV and 2.5 eV at 5.9 keV. In this paper, a detailed study of its impedance and noise is presented. The noise is not explained by a sum of known sources. The magnitude of unexplained noise is largest when the sensitivity of the TES on temperature (α) and on current (β) are the highest. The observed relation between the noise level and sensitivity suggests a source of thermal fluctuations inside the TES or between the TES and the absorber. We also found that β is linearly correlated to the product of α and current, which limits the effective sensitivity that is expressed as α/(1+β).